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Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control

Technique

January 17, 2012

Speaker: Ittidej MoonmangmeeNo.6 Student ID: 5317500117

A paper from Multtopic Conference, 2006. INMIC’ 06. IEEE

By Nadeem Qaiser, Naeem Iqbal, and Naeem Qaiser

Dept. of Electrical Engineering, Dept. of Computer Science and Information technology, PIEAS

Islamabad, Pakistan

Key references for this presentation2/18

Textbook:

[3] Bongsob Song and J. Karl Hedrick,

Dynamic Surface Control of Uncertain Nonlinear Systems: An LMI Approach, Springer, New York, 2011.

Proceedings:

[1] M.W. Spong, P. Corke, and R. Lozano, “Nonlinear

Control of the Inertia Wheel Pendulum”, Automatica, 2000.PhD Thesis:

[2] Reza Olfati-Saber, Nonlinear Control of

Undeactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles, MIT, PhD Thesis 2001.

Engineers & Mathematicians

Control Theory (or Mathematical Control Theory or Control Sciences)

Classifications & Styles of Control Paper

Control Application (or Control

Engineering)

Dynamic model Controller and/or observer design Experimental setup Simulations vs. experimental results

Engineers

Mathematicians are in a majority

Type 2: Problem formulation & Assumptions Mathematical proofs (rigorously):

definition, lemma, proposition, theorem, corollary, etc.

No experiments Illustrated examples Sometimes has no simulations

3/18

Type 1: Dynamic model (a

benchmark) Controller and/or observer

design

Computer simulation via compare with other methods

Control Sciences

Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control

TechniqueControl Theory (or Mathematical Control Theory or Control Sciences)

4/18

Outline

Underactuated Mechanical

Systems

Overview of Control System

Design

Dynamic Model

Controller Design

Stability Analysis

Simulation Results

Concluding Remarks

5/18

1q

2q

2q

1q

2q

1q

Pendubot

Acrobot

Rotary Prismatic System

1q

2q

1q

2q

1q

2q

Inverted Pendulum

Rotational InvertedPendulum

Perpendicular RotationalInverted Pendulum

“Fish Robot”[Mason and Burdick, 2000]

3q

4q

1q2q

Fully actuated: #Control I/P = #DOF.Underactuated: #Control I/P < #DOF.

E

X

A

M

P

L

E

S

Underactuated Mechanical Systems

6/18

Underactuated Mechanical Systems7/18

E

X

A

M

P

L

E

S

The Inertia-Wheel Pendulum

q1

q2I2

I1, L1g

[Spong et al, 2000]

8/18

Single-Input-Single-Output (SISO) Nonlinear time-invariant Underactuated mechanical system Simple mechanical system

Euler-Lagrange (EL) equations of motion

Control System Architecture

Step 1:

Step 2:

Step 3:

9/18

{

11 12 1 1 1 2 1 1

21 22 2

( )( ) ( )

2 211 1 1 2 1 1 2

12 21 2

11 1 12

( ) sin( ) 0

0 1

are constants

:

Q qM q g q

m m q ml mL g q

m m q

where m ml mL I I

and m m I

m q m

té ùé ù é ù éù- +ê úê ú ê ú êú+ =ê úê ú ê ú êúê úê ú ê ú êúë ûë û ë û ëû

= + + +

= =

+W

&&

&&14444244443 14444444444244444444443

&& &2 1 1 2 1 1

21 1 22 2

( ) sin( ) 0q ml mL g q

m q m q t

ìï - + =ïíï + =ïî

&

&& &&

q1

q2I2

I1, L1g

Dynamic Model

Step 1:

10/18

1 1 2 1 3 2 4 2

1 2

2

3 4

4

, , ,

.....:

.....

Let x q x q x q and x q

x x

xStateequation

x x

x

t

t

= = = =ìï =ïïï = +ïïíï =ïïï = +ïïî

& &

&

&

&

&

Collocated Partial Feedback Linearization

11 12 1 1

21 22 2 2

( ) ( ) ( , ) 0:

( ) ( ) ( , )

m q m q q h q q

m q m q q h q q t

é ùé ù é ù é ùê úê ú ê ú ê úW + =ê úê ú ê ú ê úê úê ú ê ú ê úë ûë û ë û ë û

&& &

&& &

General Form of the EL Equations of Motion for an Underactuated Mechanical System: [Spong et al, 2000]

Remarks:

fully linearized system (using a change of control) Impossible partially linearized system (q2 transform into a double integrator) Possible after that, the new control u appears in the both (q1, p1) & (q2, p2) subsystems this procedure is called collocated partial linearization

Proposition There exists a global invertible change of control in the form

where

such that the dynamics of transformedto the partially linearized system.

( ) ( , )q u q qt a b= + &

122 21 11 12

12 21 11 1

( ) ( ) ( ) ( ) ( )

( , ) ( , ) ( ) ( ) ( , )

q m q m q m q m q

q q h q q m q m q h q q

a

b

-

-

= -

= -& & &( )

1 2

Configuration vector:

( ) underactuated coordinates

( ) actuated coordinates

[ , ] ,T n m m

n m

m

q q q -

-

= Î ´¡ ¡

Control vector:

( ) controlsm mt Î ¡

11/18

Collocated Partial Feedback Linearization

1 11 1

1 0 0

new2 2

2 22

( , )( , ) ( )

:

( , )

q pq p nonlinear subsystem

p f q p g q

q pq p linear subsystem

p

ì üï ï=ï ïï ýï ï= +ï ïï þW í üï ï=ï ïï ýï ï=ï ïï þî

&

&

&

&

u

u

where (q) is an m m positive definite symmetric matrix and

10 11 12( ) (1) ( )g q m m q-= -

Define new state variables

Step 2:Transform to the Partial Feedback Linearization form

211 22 21 11

21 11 1 1 2 1 1

( , ) ( ) /( , ) ( ) where

( ) ( / )( ) sin( )

q q m m m mq q u q

q m m ml mL g q

at a b

b

ìï = -ïï= + íï = +ïïî

&&

1 11 1 12 2

2 1

3 2

(pendulum angle)

(wheel velocity)

z m q m q

z q

z q

ìï = +ïïï =íïï =ïïî

& &

&

New state equation in the Strict Feedback Form

( ) }1 1 1 2 1

122 1

11 113

3

2sin( ) ( )

(1

)

Nonlinaer Coreor Reduced

Linear orOu

z ml mL g

mz z

m

z

zer

uztm

ìï = +ïïï üï ïï ïí = - ïïï ýïï ïï ï=ï ïïþïî

&

&

&

12/18

Controller Design13/18

Goal:Stabilizes 2 30, 0z z® ®

( ) }12

2 111 1

2

3

1 2

3

1 1 1

1

1

sin( )

&

&

&

mz z

Ou

z ml z

z

mL g Core

term mz u

üïï

ìïïïïïïí = - ïïýïï= ïïþ

ïï

î

= +

ïïïï

212

(positive definite)

(negative definite)

( ) 0

( ) 0i i

i i i

V z z

V z zz

= >

Þ = <& &

First Design the synthetic inputs z2d for thecore subsystem achieves the Lyapunov stability

Core Subsystem Controller Design

Step 3:

1 2 2dS z z= -

Second define the sliding surface

121 2 2 1 3 2

11 11

1d d

mS z z z z z

m m= - = - -& & & &

2 3 3dS z z= -

2 3 3 3d dS z z u z= - = -& & & &

Third Design again the synthetic I/P, z3d

To achieve this condition, we choose

113 1 1 1 2

12 11

1d d

mz K S z z

m m

æ ö÷ç ÷= + -ç ÷ç ÷÷çè ø&

Finally, the control law chosen to driveS2 0

3 2 2du z K S= -&

To achieve this condition, we choose1

2 1 2tan ( ), 0 , 0

dz a cz a cp-= - < £ >

Outer Subsystem Controller Design

Stability Results

{ 1 2 1

1 1 1 12 11 2

2 2 2

: sin( )

( / ):

N d

L

z z S

S K S m m S

S K S

S = +ìï = - -ïS íï = -ïî

&&

&

Theorem 1:

(N , L) is global asymptotic stability

1 1 1 12 11 2

2 2 2

( / ):

L

S K S m m S

S K S

ìï = - -ïS íï = -ïî

&

&

Theorem 3:

L is global asymptotic stability

L is global exponential stability

Proposition 1: N |S1 = 0 is globally Lipschitz.

{1

11 10

: sin( tan ( ))N S

z a cz-

=S =&

Theorem 2: N |S1 = 0 if 0 < a ≤ /2 and c > 0 then z1 = 0 is global asymptotic stability.

14/18

Remark: we left out all of the proofs from the presentation

Plant parameters:

m11 = 4.83 10-3

m12 = m21 = m22 = 32 10-6

w = 379.26 10-3

Controller parameters:

a = /2, c = 9,K1 = 4, K2 = 6,

and T = 0.001

q1(0) =

I2

I1, L1

g

Initial state:

(q1(0), q2(0)) = (, 0)

q1(T) = 0g

I2

q2(T) = 0

I1, L1

Final state:

(q1(T), q2(T)) = (0, 0)

where the plant parameters are settedas same as in Olfati-Saber (2001) andSpong (2000).

Simulation Results15/18

Pendulum angle, velocity

Wheel velocity

time (second)

time (second)

2.2 sec

3 sec

time (second)

time (second)

Pendulum angle, velocity

Wheel velocity

3.6 sec

3.7 sec

[Olfa

ti-Sab

er, 2

001]

VS

Simulation ResultsM

SS

Con

troll

er

16/18

[Olfati-Saber, 2001]

VS

time (second)

Control effort (Nm)

Simulation Results

time (second)

Control effort (Nm)

0.43 Nm 0.33 Nm

MSS Controller

17/18

Concluding Remarks18/18

The collocated partial feedback linearization was presented for transform a nonlinear underactuated mechanical system into the strict feedback form

A Multiple Sliding Surface controller is designed to

achieves global asymptotically stable of the pendulum angle and the wheel velocity (neglect the wheel angle)

The MSS has advantages that the two controllers, i.e. no supervisory switching required as in Spong’s

design(2000) (more simple structure)

the response is faster than the designs by Olfati-Saber (2001) (more better performance)

However, more control effort required for MSS

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Thank youPlease comments and suggests!